dgebrd(3S)

NAME
     DGEBRD - reduce a general real M-by-N matrix A to upper or lower
     bidiagonal form B by an orthogonal transformation

SYNOPSIS
     SUBROUTINE DGEBRD( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, LWORK, INFO )

         INTEGER        INFO, LDA, LWORK, M, N

         DOUBLE         PRECISION A( LDA, * ), D( * ), E( * ), TAUP( * ),
                        TAUQ( * ), WORK( * )

IMPLEMENTATION
     These routines are part of the SCSL Scientific Library and can be loaded
     using either the -lscs or the -lscs_mp option.  The -lscs_mp option
     directs the linker to use the multi-processor version of the library.

     When linking to SCSL with -lscs or -lscs_mp, the default integer size is
     4 bytes (32 bits). Another version of SCSL is available in which integers
     are 8 bytes (64 bits).  This version allows the user access to larger
     memory sizes and helps when porting legacy Cray codes.  It can be loaded
     by using the -lscs_i8 option or the -lscs_i8_mp option. A program may use
     only one of the two versions; 4-byte integer and 8-byte integer library
     calls cannot be mixed.

PURPOSE
     DGEBRD reduces a general real M-by-N matrix A to upper or lower
     bidiagonal form B by an orthogonal transformation: Q**T * A * P = B. If m
     >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal.


ARGUMENTS
     M       (input) INTEGER
             The number of rows in the matrix A.  M >= 0.

     N       (input) INTEGER
             The number of columns in the matrix A.  N >= 0.

     A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
             On entry, the M-by-N general matrix to be reduced.  On exit, if m
             >= n, the diagonal and the first superdiagonal are overwritten
             with the upper bidiagonal matrix B; the elements below the
             diagonal, with the array TAUQ, represent the orthogonal matrix Q
             as a product of elementary reflectors, and the elements above the
             first superdiagonal, with the array TAUP, represent the
             orthogonal matrix P as a product of elementary reflectors; if m <
             n, the diagonal and the first subdiagonal are overwritten with
             the lower bidiagonal matrix B; the elements below the first
             subdiagonal, with the array TAUQ, represent the orthogonal matrix
             Q as a product of elementary reflectors, and the elements above
             the diagonal, with the array TAUP, represent the orthogonal
             matrix P as a product of elementary reflectors.  See Further
             Details.  LDA     (input) INTEGER The leading dimension of the
             array A.  LDA >= max(1,M).

     D       (output) DOUBLE PRECISION array, dimension (min(M,N))
             The diagonal elements of the bidiagonal matrix B:  D(i) = A(i,i).

     E       (output) DOUBLE PRECISION array, dimension (min(M,N)-1)
             The off-diagonal elements of the bidiagonal matrix B:  if m >= n,
             E(i) = A(i,i+1) for i = 1,2,...,n-1; if m < n, E(i) = A(i+1,i)
             for i = 1,2,...,m-1.

     TAUQ    (output) DOUBLE PRECISION array dimension (min(M,N))
             The scalar factors of the elementary reflectors which represent
             the orthogonal matrix Q. See Further Details.  TAUP    (output)
             DOUBLE PRECISION array, dimension (min(M,N)) The scalar factors
             of the elementary reflectors which represent the orthogonal
             matrix P. See Further Details.  WORK    (workspace/output) DOUBLE
             PRECISION array, dimension (LWORK) On exit, if INFO = 0, WORK(1)
             returns the optimal LWORK.

     LWORK   (input) INTEGER
             The length of the array WORK.  LWORK >= max(1,M,N).  For optimum
             performance LWORK >= (M+N)*NB, where NB is the optimal blocksize.

             If LWORK = -1, then a workspace query is assumed; the routine
             only calculates the optimal size of the WORK array, returns this
             value as the first entry of the WORK array, and no error message
             related to LWORK is issued by XERBLA.

     INFO    (output) INTEGER
             = 0:  successful exit
             < 0:  if INFO = -i, the i-th argument had an illegal value.

FURTHER DETAILS
     The matrices Q and P are represented as products of elementary
     reflectors:

     If m >= n,

        Q = H(1) H(2) . . . H(n)  and  P = G(1) G(2) . . . G(n-1)

     Each H(i) and G(i) has the form:

        H(i) = I - tauq * v * v'  and G(i) = I - taup * u * u'

     where tauq and taup are real scalars, and v and u are real vectors;
     v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in A(i+1:m,i);
     u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in A(i,i+2:n);
     tauq is stored in TAUQ(i) and taup in TAUP(i).

     If m < n,

        Q = H(1) H(2) . . . H(m-1)  and  P = G(1) G(2) . . . G(m)

     Each H(i) and G(i) has the form:

        H(i) = I - tauq * v * v'  and G(i) = I - taup * u * u'

     where tauq and taup are real scalars, and v and u are real vectors;
     v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in A(i+2:m,i);
     u(1:i-1) = 0, u(i) = 1, and u(i+1:n) is stored on exit in A(i,i+1:n);
     tauq is stored in TAUQ(i) and taup in TAUP(i).

     The contents of A on exit are illustrated by the following examples:

     m = 6 and n = 5 (m > n):          m = 5 and n = 6 (m < n):

       (  d   e   u1  u1  u1 )           (  d   u1  u1  u1  u1  u1 )
       (  v1  d   e   u2  u2 )           (  e   d   u2  u2  u2  u2 )
       (  v1  v2  d   e   u3 )           (  v1  e   d   u3  u3  u3 )
       (  v1  v2  v3  d   e  )           (  v1  v2  e   d   u4  u4 )
       (  v1  v2  v3  v4  d  )           (  v1  v2  v3  e   d   u5 )
       (  v1  v2  v3  v4  v5 )

     where d and e denote diagonal and off-diagonal elements of B, vi denotes
     an element of the vector defining H(i), and ui an element of the vector
     defining G(i).


SEE ALSO
     INTRO_LAPACK(3S), INTRO_SCSL(3S)

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